Approximating Average Distortion of Embeddings into Line
Approximating Average Distortion of Embeddings into Line Kedar Dhamdhere Carnegie Mellon University
Joint work with Anupam Gupta, R. Ravi
Finite metric spaces (V, d) is a finite metric space if V is a finite set of points. The distance function d : V × V → R+ that satisfies:
d(x, x) = 0 d(x, y) = d(y, x) d(x, y) ≤ d(x, z) + d(y, z)
symmetry 4 inequality
for all x, y, z ∈ V . Synonymous with Graph G = (V, E) with edge lengths. Distance given by shortest paths.
Low Distortion Embeddings (V, d) and (V 0 , d0 ): Finite metric spaces. A non-contracting embedding is a map f : V → V 0 . For any x, y ∈ V , d(x, y) ≤ d0 (f (x), f (y)) ≤ c · d(x, y) Parameter c is called distortion.
Average Distortion (V, d) and (V 0 , d0 ): finite metric spaces. Average Distance:
1 X d(x, y) av(d) = 2 n x,y∈V X 1 av(d0 ) = 2 d0 (f (x), f (y)) n x,y∈V Average Distortion [Rab03]: av(d0 )/av(d)
Average Distortion into line Introduced by Rabinovich [Rab03]. Related to sparsest cut. For a contracting embedding into line O(1) bound on average distortion of planar graphs. O(log r) distortion for treewidth-r graphs.
Our model Given finite metric (V, d). Host metric is the line (l11 , d0 ). Differences: Non-contracting d0 (x, y) ≥ d(x, y) (for all x, y ∈ V ). Find an embedding with minimum average distortion.
Simple lower bound on distortion Consider a star on n nodes 1
1 1
1
1
1 1
To embed into a line, in a non-contracting way Distortion = Ω(n), Average distortion = Ω(n) Any embedding has distortion O(n)
Simple lower bound on distortion Consider a star on n nodes 1
1 1
1
1
1 1
To embed into a line, in a non-contracting way Distortion = Ω(n), Average distortion = Ω(n) Any embedding has distortion O(n) Note: Average distortion can be as high as Ω(n). Yet O(1)-approximation for average distorion.
Absolute vs Relative Bounds Absolute Bounds Best guarantee about “worst case” distortion. Guarantee on distortion is independent of input metric.
Absolute vs Relative Bounds Absolute Bounds Best guarantee about “worst case” distortion. Guarantee on distortion is independent of input metric. Relative Bound Given, as input, a finite metric, embed it into the host metric to (approximately) minimize distortion. [cf. Ravi’s Talk] Comparing against the best possible distortion for the given input metric. Note: Absolute bound ρ ⇒ Relative bound ρ.
Relative Bounds: Existing Work [LLR95] minimizing maximum distortion of embedding arbitrary finite metrics into l2 via Semi-Definite Programming. ⇒ 1-approximation for maximum distortion problem. [WLB+ 98] PTAS for minimum routing cost spanning tree. ⇒ (1 + )-approximation for average distortion of embedding arbitrary (graph) metrics into spanning tree metrics. Open: Can one give an algorithm with o(log n) relative (average) distortion for embeddings into l1 ?
Our results Given a finite metric, embed it into a line in non-contracting fashion.
O(1)-approximation for average distortion of
embedding a general metric into line.
Better bounds for when the input is a tree metric.
(1 + )-approximation in time n
O( log n )
. Polynomial-time exact algorithm for tree-edge average distortion.
Warm-up: Embedding into trees Lower bound Let star(x) =
P
y∈V
d(x, y)
n2 · av(d) =
X
star(x)
x
Let m be the point which has minimum star(·) value.
1 av(d) ≥ · star(m) n
Warm-up: Embedding into trees Recall: m = argminx {star(x)}
1 X av(d) = 2 d(x, y) n x,y 1 X d(x, m) + d(m, y) ≤ 2 n x,y 2 · star(m) = n Theorem The shortest path tree rooted at m is a 2-approximation.
Getting a path Remember: we wanted a line (path) metric, not any shortest path tree. Tree could look like: 1
1 1
1
1
1 1
Getting a path Remember: we wanted a line (path) metric, not any shortest path tree. Tree could look like: 1
1 1
1
1
1 1
k -spiders: A tree with degree atmost two for all vertices except one, for which it could be upto k .
Embedding into k-spider k -repairman tour: Given k repairmen starting at a depot s. The k repairmen are to visit n customers
in a metric space. The latency of a customer is her waiting time.
k -repairman(x): the sum of latencies of all customers in a minimum k -repairman tour rooted at x.
Lower bound for k-spider From a k -spider embedding, we can construct a k -repairman tour.
x
k -repairman(x) ≤
P
y
dk (x, y)
Lower bound for k-spider Adding up . . .
1 X k -repairman(x) av(dk ) ≥ 2 n x Let m be the point with minimum k -repairman(·) value.
1 av(dk ) ≥ ( ) · k -repairman(m) n
k-spiders Upper bound (same as before)
2 av(d) ≤ · k-repairman(mx) n Theorem The best k -repairman tour rooted at m gives a 2-approximation for the average distortion of embeddings into k -spiders. Theorem A ρ-approximation for k -repairman gives 2ρ-approximation for average distortion. Currently ρ = 6 due to [CGRT 03].
Average distortion for line Fact: A line is a 2-spider. Theorem There is an O(1)-approximation algorithm for average distortion of embedding a finite metric into line.
Our results Given a finite metric, embed it into a line in non-contracting fashion. O(1) approximation for average distortion of embedding a general metric into line. Better bounds for when the input is a tree metric.
(1 + )-approximation in time n
O( log n )
. Polynomial-time exact algorithm for tree-edge average distortion.
QPTAS for trees Idea based on QPTAS for minimum latency [AK99]. Fact [AK99]: There exists a (1 + )-approximate minimum-latency tour that is a concatenation of O( log n ) TSP tours. We extend this idea for average distortion.
QPTAS for trees Divide OPT embedding into k (≈ log n/) segments.
(1 + )k−i vertices assigned to segment i. Replace the embedding of each segment by an induced “TSP-like” path without increasing the distortion too much.
(1 + )k
(1 + )k−1
...
Proof Idea Divide the objective av(d) among the segments. Share of segment i can be written as
αi · Latency(i) + βi · TSP(i) where Latency(i) = Total Latency of segment i TSP(i) = Length of the embedding of segment i
Proof Idea Cost share of segment i can be written as
αi · Latency(i) + βi · TSP(i) (Variant of [AK99]): Each segment itself can be modified to be a concatenation of O( log n ) TSP tours. This increases the distortion only by 1 + . Theorem There is a near-optimal embedding that is a log2 n concatenation of O( 2 ) TSP tours. Final Result: Can reduce the
log2 n O( 2 )
to O( log2n ) TSP tours.
Solution computed by Dynamic Programming.
Our results Given a finite metric, embed it into a line in non-contracting fashion. O(1) approximation for average distortion of embedding a general metric into line. Better bounds for when the input is a tree metric.
(1 + )-approximation in time n
O( log n )
. Polynomial-time exact algorithm for tree-edge average distortion.
Polynomial Time Algorithm P
Tree-edge Distortion: avT (d) = e∈T d(e) Theorem There is a polynomial time algorithm that minimizes average tree-edge distortion. Main Idea: The best embedding is an Eulerian tour truncated at an appropriately defined centroid. This Eulerian tour can be found efficiently. Similar to Minimum Linear Arrangement of trees ([Shi79, Chu84]).
Main Idea Local interchanges reduce average tree-edge distortion. u
v∗
u T1
T2
v∗
T1
T2
c
1.
a
d a
b
c
d
b
u
u v∗
T1
v∗ T1
T2
T2
2.
a
d b
a
c
v ∗ is the centroid of the tree.
b
c
d
Open Questions PTAS for average distortion for embedding arbitray metrics into line? Tree metriics into line? Approximating the maximum distortion of embedding a (simple) metric space (e.g. trees) into line or l1 ?
References [AK99]
Sanjeev Arora and George Karakostas. Approximation schemes for minimum latency problems. In Proceedings of the 31st Annual ACM Symposium on the Theory of Computing (STOC), pages 688–693, 1999.
[Chu84]
F. R. K. Chung. On optimal linear arrangements of trees. Computers and Mathematics with Applications, 10, 1984.
[LLR95]
Nathan Linial, Eran London, and Yuri Rabinovich. The geometry of graphs and some of its algorithmic applications. Combinatorica, 15:215–245, 1995.
[Rab03]
Y. Rabinovich. On average distortion of embedding metrics into l1 and into the line. In 35th Annual ACM Symposium on the Theory of Computing (STOC), 2003.
27-1
[Shi79]
Yossi Shiloach. A minimum linear arrangement algorithm for undirected trees. SIAM J. C OMPUT., 8(1), February 1979.
[WLB+ 98] Bang Ye Wu, Giuseppe Lancia, Vineet Bafna, Kun-Mao Chao, R. Ravi, and Chuan Yi Tan. A polynomial time approximation scheme for minimum routing cost spanning trees. In Symposium on Discrete Algorithms, pages 21–32, 1998.
27-1
Joint work with Anupam Gupta, R. Ravi
Finite metric spaces (V, d) is a finite metric space if V is a finite set of points. The distance function d : V × V → R+ that satisfies:
d(x, x) = 0 d(x, y) = d(y, x) d(x, y) ≤ d(x, z) + d(y, z)
symmetry 4 inequality
for all x, y, z ∈ V . Synonymous with Graph G = (V, E) with edge lengths. Distance given by shortest paths.
Low Distortion Embeddings (V, d) and (V 0 , d0 ): Finite metric spaces. A non-contracting embedding is a map f : V → V 0 . For any x, y ∈ V , d(x, y) ≤ d0 (f (x), f (y)) ≤ c · d(x, y) Parameter c is called distortion.
Average Distortion (V, d) and (V 0 , d0 ): finite metric spaces. Average Distance:
1 X d(x, y) av(d) = 2 n x,y∈V X 1 av(d0 ) = 2 d0 (f (x), f (y)) n x,y∈V Average Distortion [Rab03]: av(d0 )/av(d)
Average Distortion into line Introduced by Rabinovich [Rab03]. Related to sparsest cut. For a contracting embedding into line O(1) bound on average distortion of planar graphs. O(log r) distortion for treewidth-r graphs.
Our model Given finite metric (V, d). Host metric is the line (l11 , d0 ). Differences: Non-contracting d0 (x, y) ≥ d(x, y) (for all x, y ∈ V ). Find an embedding with minimum average distortion.
Simple lower bound on distortion Consider a star on n nodes 1
1 1
1
1
1 1
To embed into a line, in a non-contracting way Distortion = Ω(n), Average distortion = Ω(n) Any embedding has distortion O(n)
Simple lower bound on distortion Consider a star on n nodes 1
1 1
1
1
1 1
To embed into a line, in a non-contracting way Distortion = Ω(n), Average distortion = Ω(n) Any embedding has distortion O(n) Note: Average distortion can be as high as Ω(n). Yet O(1)-approximation for average distorion.
Absolute vs Relative Bounds Absolute Bounds Best guarantee about “worst case” distortion. Guarantee on distortion is independent of input metric.
Absolute vs Relative Bounds Absolute Bounds Best guarantee about “worst case” distortion. Guarantee on distortion is independent of input metric. Relative Bound Given, as input, a finite metric, embed it into the host metric to (approximately) minimize distortion. [cf. Ravi’s Talk] Comparing against the best possible distortion for the given input metric. Note: Absolute bound ρ ⇒ Relative bound ρ.
Relative Bounds: Existing Work [LLR95] minimizing maximum distortion of embedding arbitrary finite metrics into l2 via Semi-Definite Programming. ⇒ 1-approximation for maximum distortion problem. [WLB+ 98] PTAS for minimum routing cost spanning tree. ⇒ (1 + )-approximation for average distortion of embedding arbitrary (graph) metrics into spanning tree metrics. Open: Can one give an algorithm with o(log n) relative (average) distortion for embeddings into l1 ?
Our results Given a finite metric, embed it into a line in non-contracting fashion.
O(1)-approximation for average distortion of
embedding a general metric into line.
Better bounds for when the input is a tree metric.
(1 + )-approximation in time n
O( log n )
. Polynomial-time exact algorithm for tree-edge average distortion.
Warm-up: Embedding into trees Lower bound Let star(x) =
P
y∈V
d(x, y)
n2 · av(d) =
X
star(x)
x
Let m be the point which has minimum star(·) value.
1 av(d) ≥ · star(m) n
Warm-up: Embedding into trees Recall: m = argminx {star(x)}
1 X av(d) = 2 d(x, y) n x,y 1 X d(x, m) + d(m, y) ≤ 2 n x,y 2 · star(m) = n Theorem The shortest path tree rooted at m is a 2-approximation.
Getting a path Remember: we wanted a line (path) metric, not any shortest path tree. Tree could look like: 1
1 1
1
1
1 1
Getting a path Remember: we wanted a line (path) metric, not any shortest path tree. Tree could look like: 1
1 1
1
1
1 1
k -spiders: A tree with degree atmost two for all vertices except one, for which it could be upto k .
Embedding into k-spider k -repairman tour: Given k repairmen starting at a depot s. The k repairmen are to visit n customers
in a metric space. The latency of a customer is her waiting time.
k -repairman(x): the sum of latencies of all customers in a minimum k -repairman tour rooted at x.
Lower bound for k-spider From a k -spider embedding, we can construct a k -repairman tour.
x
k -repairman(x) ≤
P
y
dk (x, y)
Lower bound for k-spider Adding up . . .
1 X k -repairman(x) av(dk ) ≥ 2 n x Let m be the point with minimum k -repairman(·) value.
1 av(dk ) ≥ ( ) · k -repairman(m) n
k-spiders Upper bound (same as before)
2 av(d) ≤ · k-repairman(mx) n Theorem The best k -repairman tour rooted at m gives a 2-approximation for the average distortion of embeddings into k -spiders. Theorem A ρ-approximation for k -repairman gives 2ρ-approximation for average distortion. Currently ρ = 6 due to [CGRT 03].
Average distortion for line Fact: A line is a 2-spider. Theorem There is an O(1)-approximation algorithm for average distortion of embedding a finite metric into line.
Our results Given a finite metric, embed it into a line in non-contracting fashion. O(1) approximation for average distortion of embedding a general metric into line. Better bounds for when the input is a tree metric.
(1 + )-approximation in time n
O( log n )
. Polynomial-time exact algorithm for tree-edge average distortion.
QPTAS for trees Idea based on QPTAS for minimum latency [AK99]. Fact [AK99]: There exists a (1 + )-approximate minimum-latency tour that is a concatenation of O( log n ) TSP tours. We extend this idea for average distortion.
QPTAS for trees Divide OPT embedding into k (≈ log n/) segments.
(1 + )k−i vertices assigned to segment i. Replace the embedding of each segment by an induced “TSP-like” path without increasing the distortion too much.
(1 + )k
(1 + )k−1
...
Proof Idea Divide the objective av(d) among the segments. Share of segment i can be written as
αi · Latency(i) + βi · TSP(i) where Latency(i) = Total Latency of segment i TSP(i) = Length of the embedding of segment i
Proof Idea Cost share of segment i can be written as
αi · Latency(i) + βi · TSP(i) (Variant of [AK99]): Each segment itself can be modified to be a concatenation of O( log n ) TSP tours. This increases the distortion only by 1 + . Theorem There is a near-optimal embedding that is a log2 n concatenation of O( 2 ) TSP tours. Final Result: Can reduce the
log2 n O( 2 )
to O( log2n ) TSP tours.
Solution computed by Dynamic Programming.
Our results Given a finite metric, embed it into a line in non-contracting fashion. O(1) approximation for average distortion of embedding a general metric into line. Better bounds for when the input is a tree metric.
(1 + )-approximation in time n
O( log n )
. Polynomial-time exact algorithm for tree-edge average distortion.
Polynomial Time Algorithm P
Tree-edge Distortion: avT (d) = e∈T d(e) Theorem There is a polynomial time algorithm that minimizes average tree-edge distortion. Main Idea: The best embedding is an Eulerian tour truncated at an appropriately defined centroid. This Eulerian tour can be found efficiently. Similar to Minimum Linear Arrangement of trees ([Shi79, Chu84]).
Main Idea Local interchanges reduce average tree-edge distortion. u
v∗
u T1
T2
v∗
T1
T2
c
1.
a
d a
b
c
d
b
u
u v∗
T1
v∗ T1
T2
T2
2.
a
d b
a
c
v ∗ is the centroid of the tree.
b
c
d
Open Questions PTAS for average distortion for embedding arbitray metrics into line? Tree metriics into line? Approximating the maximum distortion of embedding a (simple) metric space (e.g. trees) into line or l1 ?
References [AK99]
Sanjeev Arora and George Karakostas. Approximation schemes for minimum latency problems. In Proceedings of the 31st Annual ACM Symposium on the Theory of Computing (STOC), pages 688–693, 1999.
[Chu84]
F. R. K. Chung. On optimal linear arrangements of trees. Computers and Mathematics with Applications, 10, 1984.
[LLR95]
Nathan Linial, Eran London, and Yuri Rabinovich. The geometry of graphs and some of its algorithmic applications. Combinatorica, 15:215–245, 1995.
[Rab03]
Y. Rabinovich. On average distortion of embedding metrics into l1 and into the line. In 35th Annual ACM Symposium on the Theory of Computing (STOC), 2003.
27-1
[Shi79]
Yossi Shiloach. A minimum linear arrangement algorithm for undirected trees. SIAM J. C OMPUT., 8(1), February 1979.
[WLB+ 98] Bang Ye Wu, Giuseppe Lancia, Vineet Bafna, Kun-Mao Chao, R. Ravi, and Chuan Yi Tan. A polynomial time approximation scheme for minimum routing cost spanning trees. In Symposium on Discrete Algorithms, pages 21–32, 1998.
27-1
